Technologies to rapidly obtain information about the structure of macromolecular complexes are needed to understand how these complexes function and, in particular, how aberrant interactions between molecules may result in human disease. Small-angle X-ray scattering is a relatively simple experimental method to obtain low-resolution information that offers advantages over methods such as X-ray crystallography and NMR. However, recovering information about the structure of a molecule from a scattering pattern is an extremely ill-posed and ill-conditioned mathematical inverse problem. In addition, evaluating mathematical models for a scattering pattern requires a computationally demanding, high dimensional integration. As a consequence of these difficulties, the current state of practice in SAXS reconstruction of molecular structure falls short of its potential as a scientific tool. The proposed research project is a multidisciplinary investigation into the inverse problem of determining structural information about complex bio-molecules from SAXS scattering patterns. In a close, well-established partnership with biochemists using SAXS in their research, a team of computational scientists, mathematicians, and statisticians will undertake a systematic investigation of the mathematical properties of the SAXS inverse problem using new statistical and mathematical tools, devise novel computational methods for computing solutions of the inverse problem for specific data sets, undertake an analysis quantifying the effects of model uncertainty, experimental error, and computational error on identifications made using SAXS data, and apply these methods to biomedically relevant experimental systems. The project is driven by specific research problems in biochemistry and structural biology, where extracting all available information from experimental data is essential to revealing low-resolution information for large macromolecules or complexes in solution. Such information advances understanding of the function of proteins and macromolecular assemblies in health and disease.